Math Curriculum Reform and Teachers

Title: Mathematics curriculum reform and teachers: Understanding the connections. By: Manouchehri, Azita, Goodman, Terry, Journal of Educational Research, 00220671, Sep/Oct98, Vol. 92, Issue 1

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MATHEMATICS CURRICULUM REFORM AND TEACHERS: UNDERSTANDING THE CONNECTIONS

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ABSTRACT
Ethnographic research was conducted to study the process of evaluation and implementation of 4 standards-based curricular materials by 66 middle school mathematics teachers at 12 different school districts over a period of 2 years. The data revealed that what teachers knew about mathematics content and innovative pedagogical practices and their personal theories about learning and teaching mathematics were the greatest influences on how they valued and implemented the programs. Moreover, the environments within which teachers worked were instrumental in their use of the materials. The problems teachers faced as they taught the curriculum included lack of sufficient time for planning, lack of conceptual understanding of mathematics concepts, inadequate knowledge base about how to bridge the gap between teaching for understanding and mastery of basic skills, and lack of professional support and progressive leadership.
Recommendations for reform in mathematics education uniformly call for an increased emphasis on meaningful experiences in mathematics and a decreased emphasis on the repeated practice of computational algorithms (National Council of Teachers of Mathematics [NCTM], 1989). The recommendations seek drastic restructuring of traditional mathematics curricula. As part of the reform movement, there is also growing interest in teaching approaches that support conceptual understanding of mathematics. Current visions for teaching mathematics include acknowledging the teacher as one who facilitates knowledge and who orchestrates conducive learning environments (NCTM, 1989, 1991, 1995). The recommendations also call for a reconceptualization of traditional methods of instruction.
In recent years, in an attempt to design curriculum models that support the visions of mathematics reform, educators have prepared standards-based textbooks. Implicit in the design of the textbooks is the assumption that new teaching materials will facilitate the shift from an algorithmic approach to teaching mathematics to a more conceptual approach (Ball & Cohen, 1996). In light of that assumption, a variety of local, state, and national efforts have been initiated to increase teachers' awareness and dissemination of such curricula (Cain, Kenney, & Schloemer, 1994; Evan & Lappan 1994; Keiser & Lambdin, 1996; Pennel, 1996; Pennell & Fireston, 1996; B. J. Reys, R. E. Reys, Barnes, Beem, & Papick, 1997; Steen & Forman, 1995).
The design and spread of curriculum materials to change instruction is not a new concept; it has a long tradition (Bruner, 1960; Dow, 1991). Intuitively, the argument that quality materials lead to quality teaching is appealing. However, history has shown that the parade of innovative materials in education does not significantly change school practice (Ball & Cohen, 1996; Powell, Farrer, & Cohen, 1985; Sarson, 1982). This has been due, in part, to the fact that the influence of teachers on their curriculum had been overlooked too often by curriculum researchers and designers (Ross, Cornett, & McCutcheon, 1992).
Recent research on teaching and teachers has provided evidence that how the mathematics curriculum is implemented depends on teachers' perceptions and images of the mathematics they teach (Brown & Cooney, 1985; Cooney, 1994; A. G. Thompson, Philip, P. W. Thompson, & Boyd, 1994). That is, teachers' knowledge of mathematics and pedagogy translate into practice through the filter of their beliefs (Cooney, 1994). Goals that a teacher considers desirable for the mathematics program, his or her role in teaching, legitimate mathematics procedures, and acceptable outcomes of instruction are also part of the teacher's conception of mathematics teaching (Swafford, 1995; Thompson, 1984, 1985, 1992). Recent research has shown consistently that teachers' content and pedagogical content knowledge influence how they teach and evaluate content (Ball, 1988a, 1988b, 1990; Brown & Borko, 1992; Carlson, 1988; Enderson, 1995; Manouchehri & Enderson, in press; Shulman, 1986; Shulman & Grossman, 1988).
The reasons for teachers' actions are complex. Although researchers who study teachers' thinking generally agree that teachers' personal theories and knowledge provide a basis for classroom practice (Peterson, 1988; Peterson & Clark, 1979; Shavelson, 1976; Shavelson & Berliner, 1988; Yinger, 1979), the nature of this relationship regarding standards-based curriculum decision making and implementation is not well developed (Ross, Cornett, & McCutcheon, 1992). With rare exceptions (Keiser & Lambdin, 1996; Pennell, 1996; Pennell & Fireston, 1996), research on teachers' interactions with innovative and standards-based curricula has not been reported. In particular, long-term studies that highlight the challenges teachers face as they enact the standards-based programs are missing.
In this study, we attempted to gain insight into the process of evaluation and implementation of four standards-based curriculum programs by 66 middle school mathematics teachers from 12 school districts in Missouri. I specifically wanted to provide a profile of those variables that appeared to enhance or impede the use of the programs by the participating middle school teachers.
Setting During the school years 1995-1997, 158 middle school teachers and administrators from 24 Missouri school districts participated in a curriculum review project (Missouri Middle School Mathematics Project-M3) funded by the National Science Foundation. The purpose of the project was to involve the participating middle school teachers (Grades 6-8) in the review and evaluation of four standards-based curricular materials programs in Missouri schools. The teachers had the opportunity to gain knowledge about, and experience with, curricular materials as they implemented several units from each program during 1 school year (Reys et al., 1997).
The four programs were (a) Mathematics in Context (1995), (b) Sixth through Eighth Mathematics (1996), (c) Connected Mathematics Program (1996), and (d) Seeing and Thinking Mathematically (1995). The programs had common philosophies framed by a constructivist perspective (Steffe, 1987, 1990; yon Glasersfeld, 1987, 1992) on learning and proposed similar changes in teachers' instructional approaches. The following list summarizes their goals:

1. The teacher acts as a facilitator of learning instead of as one who imparts information; he or she asks questions, probes student understanding, and encourages active learning.
2. Mathematics is meaningful to students. Students play an active role in deciding what to do and how to do it.
3. Students explore a broad range of real-life problems and make real-world applications appropriate to their level of development.
4. Students complete work instead of discrete exercises.
5. Students are introduced to computational procedures, as needed.
6. Students reflect on their work orally and in writing, asking themselves "why" and "how" questions.
7. Students work together to solve problems and to evaluate their individual and collective work.
8. Assessment is integrated into instruction and focuses on what students understand and can do, rather than what they do not know or cannot do.
9. Students and teachers share a common understanding of interpretive standards for evaluating work that includes consideration of the quality of students' understanding of task, their approaches to problems and the procedures used for solving them, their reasoning about why choices were made; the connections they make across ideas and tasks, and their communication of ideas through mathematical terms and representations.

Teaching tools such as manipulatives and calculators were used in all four programs. With the exception of one program, teaching editions for various units were in pilot and incomplete forms.
Participating teachers received a 2-day inservice training on each of the programs prior to using the materials in their classrooms. Student and teacher editions were also provided for the teachers. Teachers shared their experiences and insights about the programs with other teachers through project-sponsored conferences that occurred quarterly and concurrent to curriculum workshops. In addition, local regional groups further assisted teachers during the review and implementation of the curriculum materials. The purpose of the regional meeting was to provide teachers with a local professional network for exchanging ideas and sharing suggestions for better implementation of the materials. The regional setting was also used to engage teachers in professional discourse on mathematics reform and its implications for practice. Each regional cohort met quarterly and shared insights into the quality and substance of the units taught, as well as into student outcomes and issues concerning the implementation of the programs. The activities of each region were coordinated by a regional leader who served as a liaison between the project directors and participating teachers.
The researchers, acting as participant observers for the 2 years of data collection, were 2 of the 6 regional leaders in the project; they were involved in all stages of planning and implementing teacher activities. They observed classrooms; attended regional and project-organized state conferences; participated in planning activities at different school districts; interviewed teachers, parents, principals, and students; and reviewed lesson plans and assessment instruments developed by teachers at two different regions.
Method In this study, we investigated the process of evaluating and implementing four standards-based curricula by 66 middle school teachers over 2 years. The specific questions guiding this study were as follows:

1. What variables seemed to enhance or impede teachers' use of materials?
2. What obstacles did teachers face when they attempted to implement new curriculum materials and alternative teaching strategies?
3. What strategies did teachers use to deal with prevailing challenges?

Because of the nature of the questions, we collected qualitative and in-depth data on the participating teachers, their classroom activities, and school conditions. We therefore selected an ethnographic method (Bodgan & Bilken, 1982; Geertz, 1983; Goetz & LeCompte, 1984) of data collection. We investigated the three questions listed previously by observing teachers' classrooms and schools, interviews, surveys, and data collected through teachers' self-reports during the regional and state meetings.
In this study, we first analyzed data from each individual school setting within our regions; then we made a cross-case pattern analysis of the individual cases. We also implemented a constant-comparative analysis (Guba & Lincoln, 1981) to combine explicit coding and analytic procedures. After we completed the data analysis for each region, we studied the pattern of actions and behaviors of teachers across schools for both regions. Data analysis and data collection occurred simultaneously; each school district was studied individually. The process continued until a cross-examination of all participating schools was completed in each region, allowing us to search for common themes and issues among data. Categorizing data as they were collected was critical to making sense of how interviews, observations, field notes, and teachers' reports supported each other.
We achieved data triangulation through the use of multiple data sources and data collection over 2 years. When the first draft of this report was completed, we asked 1 teacher from each participating school district to review the researchers' analysis and to comment on areas of disagreement.
Participants
The participants in this study were 66 middle school mathematics teachers from 12 rural, suburban, and urban school districts in Missouri. Student population in each school district ranged from 500 to 4,000. With the exception of two districts, each middle school had teams of at least 2 teachers participating in the project. The school districts with more than two middle schools had between 6 and 9 teacher representatives involved in the project. The team members from each district taught different grade levels.
Among the 66 teachers, 24 taught mathematics in 6th grade, 23 in 7th grade, and 19 in 8th grade. All teacher participants had at least a bachelor's degree in education; 60 had majored in mathematics. Twelve of the teacher participants taught physical sciences, social studies, or language arts in conjunction with mathematics. The teaching experience of the participants ranged from 2 years to over 26 years. Twenty of the teachers had taught in elementary school, 8 had taught in high school, and 2 teachers had taught at the college level prior to teaching middle-level mathematics.
The teachers demonstrated some differences in their professional development experiences and backgrounds. Forty teachers were involved in at least one state professional development program within the last 5 years of their teaching experience. All the teachers claimed familiarity with the National Council of Teachers of Mathematics (NCTM) standards documents; only 10 teachers had read them. Although all the teachers reported an affiliation with professional organizations, only 6 teachers read professional journals. All the teachers reported having experience with authentic instruction, but they did not provide the nature of such instruction. Moreover, the 66 participants claimed to routinely use a variety of innovative instructional practices in their classrooms, including group work, students' self-directed investigations, and authentic assessment. All the participants had volunteered to join the project primarily to learn about the new programs. Some teachers had explicitly indicated that they hoped the project would also offer them the opportunity to learn about "how children learn mathematics" and "innovative strategies for teaching mathematics in their classrooms."
Procedure
Data collection methods for this study included observations of teachers' classroom instruction; field notes on regional meetings and state conferences; researchers' logs and field notes; individual and group surveys; and unstructured interviews with participating teachers, parents, principals, and district mathematics coordinators.
Although a majority of the classroom observations were made as a result of invitations by individual teachers, in several instances the mathematics facilitator of the school district or the team leaders from participating schools invited the researchers to visit specific classrooms. The visits were not evaluative; in all cases the researchers supported and encouraged the teachers. Some teachers asked the researchers to participate in team teaching or model teaching of specific activities from different programs. Occasionally, the researchers asked specific teachers if they allowed observations of their classes. During the visits, the researchers collected additional data on those teachers who had expressed either great success or discomfort with the study materials during the regional meetings that occurred 1 month following the first project workshop.
Unstructured interviews were conducted with all participating teachers either in person or through electronic mail. Nearly all teacher participants were interviewed twice throughout the 2 years of data collection to trace developing viewpoints and changing perspectives on both their judgment of the programs and their concern regarding the curriculum materials. The interview questions provided a thorough understanding of teachers' practices and the sources of their pedagogical decision making concerning the curricula, along with their implementation in class. The interviews were conducted primarily to gain a deeper understanding of the observations we had made of the participants' teaching and the source of their decisions. For instance, if a teacher appeared to have difficulty responding to students' questions concerning a curricular investigation, we attempted to capture the source of the teacher's difficulty through conversations and dialogue, either individually or in groups, during our regional meetings.
Throughout the data collection phase, either the teachers or the district mathematics facilitators contacted the researchers for consultation on problems that emerged as the result of using the new materials. Many of the beginning teachers asked us to validate their classroom work or to provide guidance on techniques for presenting specific topics or for using particular activities. We used the contacts to collect additional data on teachers and their schools.
Initial Data From Classroom Observations
On the basis of our observations of the participants' classroom instruction and personal interviews on specific aspects of their teaching during the first 6 months of data collection, we detected a lag between many teachers' claimed pedagogical practices and their actual instructional methods. Although on the initial project survey all teachers had claimed to use collaborative group work and authentic assessment in their instruction, our observations revealed that in over 35 cases the predominant instructional practices included using the traditional paper and pencil and individual seatwork. The teachers used group work to help students complete in-class and homework assignments, but the grouping did not always lead to collaborative learning. For instance, students were asked to complete worksheets assigned to them as a group or to share solutions to a problem or exercise. Then usually 1 student in the group read the answer(s) while the others checked their solutions. The group activities did not revolve around negotiation of solution methods or problem-solving strategies. The students did not have specific responsibilities in groups. The teacher generally did not provide systematic feedback on students' activities or monitor their progress. Moreover, assessment practices beyond the paper-and-pencil testing were rare. The test items were standard multiple choice; short essay questions were derived directly from traditional textbooks. In addition, lectures on algorithms and mathematical ideas constituted the predominant instructional practice.
The researchers categorized the participating teachers into four groups: (a) experienced (more than 6 years of teaching) and traditional (n = 19); (b) experienced, with strong ties to student-centered and conceptual (Thompson et al., 1994) instruction (n = 17); (c) beginning teachers (less than 6 years of experience) with strong constructivist perceptions on learning (n = 20); and (d) beginning teachers with traditional (algorithmic) orientation to teaching (n = 16). We created the categories to examine carefully the teachers' interactions with the standards-based curricula over the entire period of data collection and to monitor how they reacted to the programs, student outcomes, and new learning-teaching conditions that resulted from the new curricula.
Teacher Categories
We categorized teachers' practices by their general pedagogical decision-making processes as they related to interactions with students, by their choice of representations of ideas, how they dealt with students' cognitive difficulties as the students encountered new ideas and problems, and by the teachers' attempts to create and sustain an inquiry-based learning environment stemming from social discourse. We used Thompson et al.'s (1994) descriptions of conceptual and algorithmic orientations to teaching to help formulate several criteria for evaluating the teachers' classroom behavior. The descriptions include:

1. The amount of time teachers allocated to lectures and discussion of algorithms. Traditional teachers consistently devoted 60-70% of their class time conducting whole-group lectures and presenting ideas and concepts in a direct teacher-dominated method. In contrast, innovative teachers spent between 50-60% of the class time on open-group investigations and problem solving and about 10-15% of the time on skills practice.
2. Amount of worksheets and individualized drill and practice exercises teachers used in class. Traditional teachers complemented their lectures with specific worksheets that emphasized particular algorithmic skills during class time. Innovative teachers, however, used worksheets that included both problems and standard exercises as homework to complement open investigations that were completed in class.
3. Types of questions posed by teachers in class. Traditional teachers generally asked low-level cognitive questions, presented in the form of dichotomous yes-no answers. Innovative teachers asked why and what if questions to extend the domain of students' mathematical activity.
4. Types of responses provided by teachers to student questions. Traditional teachers provided answers to all the students' questions; they were the authority in resolving issues during instruction. The innovative teachers opened the questions to the whole group and allowed all the students to respond. They usually provided students with hints, rather than correct answers, to encourage students to consider problems, to make conjectures, and to test them.
5. Reactions to groups when pupils were stuck on a problem. The traditional teachers felt responsible for clarifying all ambiguities about the problems posed in class. They answered all questions and assumed full responsibility for clarifying critical points within the task. The innovative teachers used students' confusion as an opportunity to extend students' independent exploration and problem solving. When faced with such situations, the students asked their peers to help clarify the problem areas.
6. Teachers' methods of monitoring small groups' activities and progress. The traditional teachers tended to randomly assign students to groups and to provide time to complete the assignments. The nature of the teachers' interactions with the small groups was more management oriented than cognitive. The innovative teachers apparently planned the groups in advance, strategically assigning students to specific teams. The responsibilities of each group member were announced based on specific strengths and weaknesses of the students. Those teachers also walked around the room and posed questions to students who did not consider problems that could lead to over- or undergeneralizations.
7. Teachers' methods of synthesizing ideas and concepts in class. Following group activities or completion of a solution to a problem, the traditional teachers asked students to offer the final solution without a planned follow-up discussion. As students completed the assignment, they did not attempt to help their peers synthesize the significant ideas embedded in the activity or to consider other solutions. In contrast, the innovative teachers emphasized the process of obtaining solutions. They invited students to agree or to disagree with one another's methods of solving problems. Also, the teachers had built time into their lesson for debriefing the students who shared ideas with their peers.
8. Teachers' methods of dealing with students' lack of understanding of concepts. The traditional teachers encouraged students to be patient and to give more examples in order to understand what caused the problem. In contrast, the innovative teachers used multiple forms of representations such as the use of manipulatives, physical models, and drawings to provide a different perspective for the learners. They also relied on other students' input and explanations when their own explanations seemed not to help learners construct a better understanding of the problem.
9. Teachers' attempts at learning about students' understandings and misunderstandings. When dealing with students' difficulties in understanding certain concepts, traditional teachers repeated the same ideas using a similar example. Students were encouraged to "do" more examples as a way of learning the procedure. In contrast, the innovative teachers attempted to understand the sources of the students' cognitive obstacles and asked for the students' explanations about what they did comprehend.
10. Teachers' lesson planning procedures. The traditional teachers had planned their week's lessons in advance. They followed specific time lines for content coverage. Their lessons originated, for the most part, from the standard textbook previously used in class. The unexpected class discussions did not change the routine of their instruction. In contrast, the innovative teachers accounted for the daily classroom events when designing their lessons. Because they anticipated potential problems as students encountered certain concepts, the teachers supplemented their lessons with descriptive activities. Those teachers spontaneously modified their lessons to accommodate unexpected and interesting mathematics discussions that arose during their instruction.

The preceding criteria were highly intertwined. In all instances, those teachers with a strong orientation toward a particular approach to teaching modeled their lesson plans on all or a majority of the 10 categories. However, for beginning teachers, the categorization was not as clear or straightforward. In at least 12 cases, the beginning teachers vacillated between the two pedagogical practices. For instance, although they could not always provide pedagogical models or multiple representations during instruction, the beginning teachers helped students find additional resources after the sessions. The beginning teachers spent less time on direct lectures, or used fewer worksheets, so they were not proficient in interpreting students' responses, listening to students' explanations, or making sense of their comments. Therefore, the teachers' choice of mathematical concepts was not diverse. In addition, they could not always connect the students' dialogue to a significant mathematical idea; however, the teachers did ask the students to explain their thinking, and they listened to the students' reasoning.
We relied on personal interviews to obtain additional data about the beginning teachers' instruction and beliefs on the learning--teaching processes. We also used the interview data to further substantiate or refute our conjectures on how we classified certain teachers under the categories identified earlier. With the additional data, we attributed the inconsistencies in performance for this group to a lack of experience rather than to a strong traditional orientation toward teaching.
Findings From the beginning, we asked the teachers to implement the projects as much as they felt was possible for their given situation. Thus, teachers varied in (a) time they spent using the materials, (b) expectations from the students, and (c) amount of effort invested in building the type of classroom culture that was conducive to students' productive use of the materials. During the regional meetings, participants reported on the amount of time they spent on curricular programs or whether they had used the classroom materials consistently. They elaborated on the classroom activities they liked, why they liked them, and what they viewed as the positive or negative impact of their use. The teachers' knowledge of content and innovative pedagogical practices appeared to be a strong influence on what and how they used the projects and evaluated them. That finding was persistent in how they viewed the effect of the materials on student learning and their evaluation of the students' work.
The frequency and intensity with which the teachers used the materials implied their success or failure to resolve the obstacles they faced during the implementation process, the professional support provided for them at the school and district levels, and their personal theories on learning and teaching. All the participants began their activities enthusiastically; however, only 20 of the 66 teacher participants reported routinely using the programs in their classes after the first 5 months of the project's conception. Nearly all the teachers reported increased student interest in learning mathematics and greater student involvement in class activities as a result of using standards-based materials. However, those teachers accustomed to student-centered and constructivist instructional practices were more enthusiastic about using the programs and evaluating their impact on their students' learning, as well as on their own practice. Moreover, in those schools in which the teachers were supported both emotionally and intellectually and where opportunities for productive growth of their content and pedagogical skills were provided, a sustained spirit of reform was easier and more natural. In contrast, in schools in which the work environment was not supported and constructive practices were not encouraged, teachers' attempts to use the standards-based programs became episodic.
Teachers' Experiences and Personal Theories
The more experience that the teachers had teaching with traditional approaches, the more they questioned the value and relevance of the programs. The traditional teachers generally questioned both the value of the mathematics content discussed in the materials and the adequacy of the suggested activities for the grade level they taught. Those teachers were concerned about covering curriculum content requirements suggested in traditional textbooks. In essence, the programs were an affront to their theories about the strategies, methods, and materials they had developed and used successfully in their classrooms and what they considered as legitimate mathematics for the grade level they taught. The most frequently cited explanations offered by that group for not using the programs consistently in their classes included lack of depth in mathematical investigations (n = 20), inconsistencies between the topics discussed in the units and what their curriculum required (n = 18), and the need for preparing students to do "real math." The following excerpts from field notes taken in February 1996 from three participating teachers during one of the regional meetings captured the perspectives offered by the group:
I don't think these activities are going to help my students get ready for algebra. Don't get me wrong, I think they are really good for younger kids. It is going to help them with their self confidence but I have my curriculum to cover. (Gay, Grade 8 teacher)
These are interesting activities to do with kids but they need to go beyond the manipulative at this point. We need to teach them abstractions. For example, the fraction strips would be good for younger kids but for students in 8th grade they need to learn the algorithms, be able to add or subtract fractions. Just working with the manipulative is not preparing them for that--they need more practice. (Janice, Grade 7 teacher)
Maybe if the students had used these programs earlier they would be ready for what we have to do with them in the 7th grade. But I have kids in my class that cannot even sit still for five minutes, they don't even know their multiplication table. I have too much to do with them to get them ready for the test at the end of the year. Doing investigations that are in the programs would not help them get ready for the test. It is not a good investment of the time. (Terry, Grade 8 teacher)
In contrast, those teachers with extensive backgrounds in nontraditional teaching found the mathematics programs to be intellectually stimulating, if not always practical. The practicality of the programs was determined by the amount of time the teachers felt they needed to work on a particular activity or to prepare classroom materials. Generally, the teachers in that group considered the program to be worthwhile as long as it offered a new way to approach the classroom. The following are comments made by teachers at regional meetings held from March through April 1996:
I like many of the lessons they suggest. In fact, I have used some variations of a lot of these activities in the past. The ideas about fraction strips is good but I much prefer using the cuisenair rods. They are easier to handle and we don't have to spend so much time cutting out paper. So, I am using stuff from my own files at this point. Next week though I like to try out their activity with building quilts. I am really curious as to how it will go with the kids. (Carey, Grade 6 teacher)
I have had a hard time in the past teaching kids about volume. They don't seem able to understand volume as well as other measurement concepts. I am going to try the lesson with Lemonade mixture and see if it helps them. For now, I have other area activities that I know they will like that I am using. (Connie, Grade 7 teacher)
I like the self-assessment component of the program. I have modified it a little and am using it with my students throughout all I do with them. I have done journal writing in the classroom where they actually write about what they learn in class but I like the self assessment better because it is so focused. (Michelle, Grade 6 teacher)
The beginning teachers, whose teacher preparation programs had emphasized constructivist approaches to teaching and learning, confidently embraced the curriculum materials. Those teachers were concerned about the affective consequences of using the programs in their classes; their judgment did not, however, include a thorough analysis of the content of the materials.
The beginning teachers were often concerned about what and how to teach the next day's lesson or a particular activity; however, they were highly committed to using the standards-based materials. That effort was greatly influenced by their students' enthusiasm for learning and what they perceived as the positive impact of the programs. The teachers also maintained a philosophical approach to the nature of the new textbooks, although they sometimes felt inadequate in facilitating classroom investigations and discourse. The following comments were excerpted from participants at regional meetings held in October 1996 and September 1997:
I cannot tell you how much my kids love the programs. They are so active in class all the time, even the ones that did not do much in class before are now either making things, putting together posters, or helping other classmates solve problems. I know this is the way we should teach mathematics, you know, getting them involved. That is what constructivist teaching is, right? (Tammy, Grade 8 teacher)
I know there are still some problems as I am doing these activities and units. Like, how to get them to write--some of my kids cannot even read at this point. I just have to keep doing it. I just do not want to teach math the way I was taught. I believe in what NCTM says and I know I should use these. (Toni, Grade 7 teacher)
I had a hard time understanding this geometry stuff. I was never good at it and this unit was on spatial skills. So, I gave them the materials and I told them to work on them in groups of three. They finished the perspective drawings so quickly I could not believe it. I keep thinking they were right when they said we need to give students things that are fun and engaging. I want to give them this kind of activities all the time. (Karin, Grade 6 teacher)
For those teachers who had limited teaching experience and were unfamiliar with instruction methods that exceeded traditional methods, their judgment of the programs and their use was based on whether ideas could translate into an activity they could use with confidence. That group was less affected by philosophical issues; their perceptions of the programs were shaped by the amount of useful materials they perceived. Those teachers used the materials either in addition to their more traditional lessons or with a particular group of children. In particular, seven of the teachers used the program activities with only special education children who were unable to handle the regular pace of the classroom instruction.
Classroom observations of teacher participants with limited experience, as well as of those with a strong traditional orientation to teaching, proved that while maintaining their routine instruction, both groups of teachers used a particular program or unit as an enhancement activity, which often lead to the production of a certain artifact for classroom decoration, without any discussion of its mathematical significance. Such superficial use of the materials did not affect the activities of the students, and, in most cases, the students found them irrelevant to their mathematics learning. That message was reflected in the teachers' reports of their students' judgment of the programs. The researcher's conversation with Pain and Lisa, two of the beginning teachers, during a school visit May 14, 1996, highlight the point clearly:
Pam: I am not going to use the units anymore because students keep asking me why they do them when the activities have nothing to do with the math they are supposed to be learning.
Lisa: I have the same problem. Just yesterday after spending so much time on activity with drawing perspectives by putting the blocks together, one of my kids asked why they were doing it since it did not have anything to do with the chapter they had worked on about fractions.
Researcher: How did you respond to their comments?
Lisa: Just that this was something fun to do and that it was also mathematical.
Researcher: How about you Pam? What did you tell your students about why they were doing the activity?
Pam: (umm...) Well, I told them that it was Friday and we wanted to do something different! I also told them that we had to do this program pilot in our classes to see how they would like the materials.
The teachers who had extensive experience teaching elementary school students easily adopted the instructional practices required by the mathematics programs. Those teachers were familiar with student-centered instruction and did not follow a particular textbook guideline. In contrast, the teachers with extensive high school teaching experience were dissatisfied with the classroom organization that the new programs imposed on their environments. They felt that the materials did not adequately prepare their students to engage in in-depth discussion of topics fundamental to their success at the grade level.
Social Influence
For teachers in this study, knowledge of teaching practices came primarily from colleagues and other school professionals. Professional colleagues helped the teachers interpret and evaluate the curriculum programs and classroom practices. To a large extent, the teachers developed their understanding of what a particular program was about and its influence on instruction and learning on the basis of the accounts and assessments of others. The long-range impact of such social influences on teachers was twofold.
On one hand, teachers whose work environments emphasized active learning and constructivist philosophy used the materials consistently; this occurred also for those teachers who had little classroom experience. In 5 of the 12 school districts in which strong ties among the faculty existed, growth, development, and commitment to consistent use of materials were most prevalent. That group was eager to learn alternative ways to conduct investigations, how materials were used by other classroom teachers, and students' perceptions of and reactions to certain activities.
In essence, the group gained additional insight into how to deal more effectively with their classroom issues by sharing experiences with a larger group. The collegiality was most critical for the beginning teachers and those with limited middle school teaching experience. The fact that the veteran teachers faced similar concerns about the new curricular materials gave the less-experienced teachers confidence to continue to try different programs and to be optimistic toward the program goals. The following excerpts from the interviews with teachers capture the powerful impact of such collegiality on beginning teachers:
When we started using these programs I was so excited. Especially after the first workshop. I had no ideas that it was going to take so much time and energy to do them with students. Sometimes it seems so much easier to just stand up and lecture and then give them worksheets. Then I look around and I see even Carey and Michelle having the same problems and I think, okay this is going to be all right--if they are struggling to make them work with all the experience that they have, then it is natural that I would too. I know I am going to do better next year. (Connie, Grade 6 teacher, April 1996)
I get really frustrated when I know some of my students can't read the investigations and therefore they cannot do them. Carol told me that what she does is asking the good readers to help the others. I started doing that in my class too and it is really helpful... William also gave me some tips for helping them write. He said that he had the same problem with his first hour group. I went and watched him teach during my planning time. They were doing great--he keeps telling me not to give up and I know he is right. So, I am learning to tough it out and I am learning so much. (Tammy, 3rd-year Grade 8 teacher, March 1997)
I gotta tell you I was really skeptical about these (pause). I never had seen mathematics like this. I am trying really hard--everyday is a new problem to deal with. If it is not the groups, or the kid that cannot read, it is the kid that cannot even add or multiply. How are they going to do problem solving when they cannot even add or multiply? In our last faculty meeting Carey showed me a couple of things that I want to try out in class--if it was not for her and Carol I would have gone back to the old stuff in no time (laughs). (Sarah, 2nd-year Grade 6 teacher, April 1997).
In the interview settings, the teachers were motivated; they demonstrated the tendency to ask general questions concerning the mathematics reform and philosophical framework of the programs. They also concentrated on the positive outcomes of the mathematics program. The beginning teachers approached curricular issues holistically and globally; they included not only individual teacher dilemmas and classroom circumstances but also general issues concerning assessment, mastery of basic skills, and heterogeneous grouping. Furthermore, the faculty met to pose problems, share insights, and solve problems. The purpose of the group was to "make the curriculum materials work" in classrooms. That was particularly evident in two of the school districts in which the mathematics facilitators had organized monthly faculty meetings to discuss the implications of using the new standards-based curricula for the teachers' instruction.
On the other hand, in schools where the teachers were surrounded by colleagues and peers who were skeptical about the standards-based curricula as well as about the practicality of the classroom practice materials, the teachers were less inclined to use the programs. The beginning teachers felt obligated to employ traditional practices; their use of the mathematics programs was not systemic and consistent. In those situations, even the teachers with constructivist perspectives on teaching and learning reverted to a traditional routine of classroom instruction. Excerpts from personal interviews with the teachers follow:
I really want to do these programs more often but I don't know if I can, They keep telling me that it is about time when we should start the standardized test practices. I know that it should not be a problem in the long run. I know what kids are doing with problem solving and writing stuff is what I should be doing, but my principal was in my classroom yesterday. This was my first observation this year. He wanted to see my lesson and when I showed him what I was going to do he asked what homework problems I was going to give them. Well, they just had to finish the investigation at home and that was my homework assignment--he did not like it though--he thought the kids needed to do more practice. So, I copied a few worksheets for them. I know this is wrong but what can I do? (Andy, Grade 6 teacher, December 1996)
Darlene and Jamie tell me that this is just a fad like it was in the 60s. That, no matter what new things come around, they will go out after a couple of years and we are back on the same track that we were before. See, I know this is not the same. Everything I have read tells me that we need to be doing this sort of thing. And, I try to talk them into using some of the activities so they would see for themselves how good they are but they don't hear of it. They are now working on the chapter on variables--they say I am falling behind and I know I am. So, I just have to go back to my old text for awhile and try to catch up. (Dana, Grade 7 teacher, January 1997)
The teachers who worked within such environments had to use traditional textbooks and instruction methods because of school administration pressure for standardized tests and evaluation. The teachers' attempts to fulfill the obligations imposed by the school and district typically lead to excessive algorithms and therefore impeded their attempts to develop a form of practice required by the standards-based curricula. When the teachers did not have a broad understanding of the role of the middle school mathematics curriculum or had not formulated a vision of what that curriculum should entail, the administration's demands served as a reason for not using the materials or for not using them consistently.
Leadership and Professional Support
The activities of the teachers, either individually or collectively, depended heavily on the leadership available to them at the school and district levels. The leadership, provided by a mathematics coordinator, principal, or expert teacher with an excellent reputation in the school district, was the primary social influence on the participating teachers. For all teacher participants, the presence or absence of progressive leadership was instrumental in their persistence to use the mathematics programs and to implement them effectively in their classrooms. The teachers frequently followed the lead of a colleague. They relied on the support and guidance of their leaders to make judgments about the programs, curriculum, and instructional practices that they used. Moreover, the teachers depended on the leaders' suggestions on the "how to" questions that constantly emerged during the implementation phase. The questions included how to organize and monitor heterogeneous grouping, design and use authentic assessment, and pace instruction in all the classes.
The impact of the leadership was most visible in the work and development of beginning teachers and those with limited classroom teaching in the middle grades. For those teachers the presence of a support system and progressive leadership generally helped them to restructure their curriculum and instruction with limited fear of the outcomes. Therefore, teachers with limited knowledge of content and curriculum could ask questions and seek ideas about how to use certain activities, how to teach a certain concept, and how to manage group activities. The following excerpt was taken from a personal interview:
I manage to help them change one or two things about their teaching--to set goals for doing new things. I think we are going to be okay. I have talked to Dave [the school principal] and explained to him several times that my teachers are working really hard as it is--if we want them to do more then we have to compensate for their time. He talks about parents being unhappy. I told him to let me deal with parents. So, we organized a family night and showed the parents that came specific examples of what teachers are doing with their children. My teachers need to be supported if they are to be successful and it is my job to make it easier for them. I can see my teachers doing things differently and I know it is not easy. I will do anything I can to support them. (Carol, mathematics facilitator, April 1996)
The next excerpt was sent by e-mail:
I know some of my teachers have difficulty using the graphing calculators in class. They are not comfortable with them and don't know how to use them. I told Linda that I would go to her class and model for her how to use them for an investigation. I know I should do more. Next month though for our professional development day all we will do is graphing calculators. This should get some of them more comfortable. (Nancy, mathematics coordinator, September 1996)
In those settings where progressive leadership existed, the leaders attempted to systematically involve all the mathematics faculty from the district's middle schools, including those not involved in the project, in the review and use of the standards-based curricula (Manouchehri & Sipes, in press). In light of possible adoption of the new curricular materials, the leaders initiated ongoing dialogue among the faculty and designed professional development opportunities to help the teachers develop the necessary skills and knowledge to change their practices.
In contrast, when placed within a work environment and pressured by the type of leadership that emphasized an algorithmic orientation to teaching, even those teachers who held strong beliefs about maintaining student-centered learning environments and teaching for understanding reverted to a traditional instruction routine. That scenario was more prevalent when not only the school principal but also the district's veteran teachers magnified the difficulties associated with the standards-based curricula, rather than emphasizing their positive outcomes on student learning.
Issues and Obstacles
The use of new curricular materials mirrored many problems that teachers faced; the obstacles were highly intertwined and difficult to separate. There was an incongruence between the teachers' perceived challenges of using the programs and the actual challenges that surfaced. On a survey conducted at the beginning of the 1st year of teacher involvement in the research, 62 of the 66 teacher participants anticipated "parents' resistance to change" as a major obstacle in implementing innovative programs in their school districts. Both veteran and beginning teachers identified parents' lack of satisfaction with a shift away from traditional textbooks and from teaching basic skills as a major hindrance to their intended classroom goals. Parent concern was an issue for many teachers in several school districts, but it did not appear to be critical in many cases.
Some parents who were involved in their children's education expressed concerns about various programs. The concerns were reinforced when parents felt unable to assist their children at home because of the nature of the assignments. The teachers routinely cited parents' expectations about using traditional materials and approaches as the main reason that the teachers did not (either fully or partially) embrace a new program. Several beginning teachers and those teachers unfamiliar with the learning and teaching procedures demanded by the new curricula were challenged by the parents. In settings where leadership support was not provided for the teachers, they used parents' lack of familiarity with the new materials to justify using more traditional practices. The following comments from a personal interview capture the essence of the concerns of the group:
They [some parents] called the principal and told him that their kids aren't bringing home assignments. They told him that their kids are not learning what they have to learn. One of the kids was an A student last year and now with all the writing and problem solving stuff, she is not doing so good. I asked the principal to invite them to come over to my class so we could talk about the situation. I tried to tell them about the NCTM recommendations and that we wanted kids to do different things than before (pause) they just did not want to hear of it. They could not understand that the reason their kids were not doing well was because I was not pushing the algorithms anymore. That there are more important things than just being able to manipulate numbers. The principal did not give me an ultimatum (laugh) but I guess I should do more worksheets and things so the parents would be happy. I am going to go back and use the old textbook too. So, the kids could take home something parents had seen before. (William, Grade 6 teacher)
On the other hand, administrators at the district level who supported the innovative changes occurring in classrooms encouraged the teachers to adopt a proactive stance and to communicate with parents the goals of mathematics reform as well as the significance of what they were teaching. The parents supported the teachers' efforts in all situations in which the parents were given a rationale that surpassed philosophical and theoretical perspectives on the use of the new materials. Conversations with parents revealed that they felt most secure when the teachers were articulate and confident about their own practice and about the mathematical value of what they taught in their classes. Consequently, dealing with parents was particularly difficult for the beginning teachers and for those less articulate about the rationale for curricular change.
Other more pressing barriers, however, were conceived after teachers began using the programs in their classrooms. Their concerns were practical, reflecting the circumstances within which they worked and various demands placed on their time and energy. Those obstacles were handled differently by teachers depending on their previous personal and professional experiences as well as on the nature of professional expectations within their working environments. The teachers' concerns originated primarily from their lack of knowledge about the long-range content goals of the programs, the way the units were organized, the extent to which ideas and topics were addressed in various units, and the new instructional practices demanded of them.
The paramount issue in this study was that although the new programs brought about numerous exciting and enriching activities, they did not provide the teachers with detailed methods of how to address the content development. Although the program materials included carefully designed investigations, they did not instruct the teachers to consider the long-term dimensions of curriculum construction and implementation. Often the teachers had to decide how to connect ideas between various activities and units and how to deal with the content at hand. Moreover, the materials did not provide the teachers with an understanding of how to deal with the students' thinking, their multiple approaches to problems, and misconceptions as they encountered different ideas. That deficiency in the programs lead to the creation of other related problems (discussed below).
Time
One of the major challenges for the teachers, regardless of the extent of their experience and familiarity with authentic instruction techniques, was the lack of adequate time for planning and instruction. Nearly all the teachers cited time as a critical issue for successfully implementing materials. It became increasingly evident to all the teachers that, with the new programs, they needed additional time to plan and organize their classes. The extra time was required to design lessons, conduct classroom activities, assess student progress, plan additional activities to help students develop the necessary skills to complete activities, and pace the teachers' instruction for groups of students.
The time lines for implementing various units were not accurate; the teachers needed much longer than anticipated to teach the materials. In those classrooms in which the students were not familiar with using manipulatives or engaging in collaborative learning activities, the teachers spent more class time familiarizing students with the new skills. Some of the teachers perceived the extra time as a hindrance to the lessons that they had to teach during their class time; those teachers therefore did not use the materials in their classes or used them only occasionally. Many teachers believed that changing classroom culture was impossible, considering the amount of time they had with students.
Over 55 of the teachers spent longer hours after school to read students' work, provide group sessions, learn to use the manipulatives suggested in the investigations, or work with parents who wanted to learn how to help their children at home. The additional hours created tension between teachers' professional and personal schedules. The issue became an even greater dilemma for 7 teachers who were involved in professional activities. Those teachers found that engaging in further collaborations with their school faculty or other project members or attending to mandates of the curriculum projects added extra demands on their schedules. That group, primarily veteran teachers and constructivist practitioners, believed that the new curricula was more of a hindrance than an enhancement to their teaching routine. They became less involved in project activities and less motivated and willing to use the standards-based curricula. For those 7 teachers, using their already successful materials became the norm; using the new programs was less systematic. That group was visibly absent from the regional- and project-sponsored state conferences. Their perceptions were forwarded to me in the following e-mail message:
I don't feel I am learning much here anymore. The things we are told are the things I have done already. It is not that I don't find value in them but perhaps they are better for the younger, less involved teachers. Being the department chair this year and doing all the new assessment standards is consuming too much of my time. Something is got to go. I guess this is it--maybe it will get better next year. (Carey, Grade 6 teacher, September 1997)
The consequences of such an extensive demand on the teachers' time was dramatic. Many of the teachers who worked in nonsupportive and isolated environments used the time demand as a rationale to either shy away from using the materials or to argue against the value and practicality of the programs. Four of the younger and less experienced teachers did not fully implement the program materials or use them consistently because of the tensions caused by the personal and professional demands imposed on them.
Basic Skills
For all teacher participants, the challenge of creating a balance between what they perceived as the necessary algorithmic knowledge for each grade level and the development of conceptual knowledge of topics was an ongoing dilemma throughout the data collection period. The major challenge (and an extremely time-consuming one) for many of the teachers was seeking ways to create an equilibrium between teaching for mastery of basic skills and teaching for understanding of mathematics concepts. The challenge was closely affiliated with the varying levels of mathematics backgrounds that children brought to the learning environment and to the teachers' classrooms. The dilemma obstructed the work of all the teachers, even those with strong constructivist perspectives on learning and a conceptual orientation to teaching.
Although several participants, particularly those in progressive work environments, maintained a philosophical outlook and agreed with mathematics reform recommendations that the two domains were not necessarily dichotomous, they also struggled to determine how to improve their students' mastery of basic skills. In nine of the school districts under constant pressure by the administration and peers, the participating teachers began using worksheets in their classrooms on a regular basis. Sixty-two of the teachers felt that the curriculum project did not adequately address the amount of "practice of the basic skills" needed. Those teachers had difficulty adjusting to the fact that curricular approaches to teaching mathematics did not assume that concepts or skills introduced in a unit would be mastered within that unit. When the teachers used the new materials for the first time, they were unsure which concepts they could build up slowly and completely for later use; this gradual procedure was new to the teachers. Consequently, they needed to supplement or to bring closure to a unit by discussing an algorithm and by using traditional worksheets to establish mastery. One of the veteran teachers articulated the concern as follows:
As much as I like to believe that through what we do they [students] will also learn the basic skills and algorithms, I am not so sure that they learn all that they are supposed to learn without some practice. (pause) God knows I am not into these drill and kill stuff but students do need to have some direct experience with the procedures so they could at least remember them later when they see them in a different context. I don't see why we should keep it a secret. (Jane, December 1996)
New Demands
In nearly all the programs, individual and group assignments as well as homework problems required considerable reading and writing. That situation posed a major challenge for both students and teachers who had little or no experience in using such techniques in mathematics classrooms. It was particularly critical for those students who read below grade level. The need for students to acquire new skills created additional responsibilities for the teachers. They had to constantly help students to read mathematics for understanding and to help them write about their mathematics solutions. The situation was problematic for nearly all teachers involved; it was a major obstacle for the traditional teachers who consistently used the materials and for beginning teachers who were not familiar with the new methods. For those teachers with strong ties to traditional teaching strategies, students' lack of immediate success in reading and writing in mathematics was a rationale for not using the activities in their classrooms. However, those teachers whose education emphasized reading and writing in mathematics helped their students develop reading and writing skills during class, in collaboration with other teachers and school resource personnel.
The veteran teachers with a strong algorithmic orientation to teaching generally dismissed the new demands as irrelevant. Students' journals from various teachers' classrooms revealed that because of their lack of practice, the students were unlikely to write coherently or to communicate their thinking processes and methods of problem solving when specific writing tasks were required. The teachers frequently used the students' inadequacies in reading and writing mathematics as a rationale to dismiss the usefulness of this component of the new programs. According to one of the teachers, the new procedure was not only "not helping the students but it is hindering their learning of what they need to learn in the mathematics classroom." Another teacher said that as a mathematics teacher she had "more important things to do than spending time doing what needs to be done in the language arts classroom."
Although reading and writing were one aspect of the programs that was constantly encouraged by beginning teachers, those teachers could not always use the students' responses for evaluation. The teachers spent a significant amount of time outside the classroom, reading and commenting on students' assignments; assigning grades to the students' work was a problem. Because the teachers were operating within a traditional grading structure and were being pressured by parents to assure the students' success, they generally used traditional assessment instruments and evaluated basic skills mastery. Therefore, the teachers experienced greater demands on their time for planning and evaluating student progress.
Content and Pedagogical Content Knowledge
To help students improve their mathematics ability, as envisioned in the new curricula, the teachers had to have a conceptual knowledge base about mathematics, along with a substantial knowledge of learners' cognition. The teachers needed to know how ideas in mathematics were connected and which ones were structurally central because the structure of the materials was such that the organization of the content was oriented to open-ended mathematical problems, rather than specific topics. Furthermore, the teachers needed to know how to build on students' insights toward mathematical investigations and how to connect those personal insights to a more sophisticated mathematical domain. The teachers had to have knowledge of students' thinking, to understand students' cognitive obstacles as they encountered different investigations, and to plan teaching methods to facilitate learning of those concepts.
In all four curricular programs, ideas and algorithms were not explicitly presented and concepts were developed through interrelated activities and investigations; therefore, the teachers needed to possess mathematical knowledge that was broad and contextual. The teachers had to "see" where the mathematics was in the activity. However, that requirement was not applicable for many of the teacher participants in this study. The teachers had widely varying amounts of substance and knowledge in their mathematics backgrounds. During regional meetings and school visits, the teachers asked us to identify the mathematics themes that were addressed in the investigations. About 30 of the teachers consistently questioned "how" and in "what ways" investigations from the same unit were related. They did not see the connections not only among the investigations but also between problems explored in the units and familiar mathematics topics. The link between specific concepts embedded in the interrelated investigations was not obvious to those teachers.
Therefore, a crucial factor in whether the materials were consistently used in classrooms and valued by the teachers depended heavily on their content background and confidence in their knowledge of mathematics. The teachers avoided those mathematics units that they did not feel comfortable teaching or those that they did not view as mathematically significant. That situation, coupled with the fact that when different programs were used, students began asking important why questions to which teachers with limited mathematics knowledge could not respond adequately, created an intense and even less secure situation for the teachers. When the teachers did not have immediate answers to the student-generated why questions, they either dismissed or ignored them. The consequence of this practice was detrimental to both teachers and students. Teachers became less secure in what they thought they knew; students were less willing to participate in activities and to ask questions. The following example from a classroom episode illustrates what we observed in many classes conducted by teachers with limited mathematics knowledge. During that session, Bonnie, an 8th-grade teacher, was asked by students to resolve conflicting results among the solution methods obtained from a problem that was a suggested warm-up activity by one of the curricula.
Bonnie assigns the problem: It is Karen's birthday and Karen's mother bakes 13 cookies. People at the party take turns picking cookies from the tray. If Karen takes the first cookie and the last, how many people were at Karen's party? Following an 8 minute work time, Bonnie asks students if they are ready to present their solution to the group. Several students raise their hands. SI explains that his answer is 12--it seems that the answer is consistent with what Bonnie considers as the correct answer. Good work S1. Does everyone agree with the solution? Bonnie asks. S2 suggests that her answer is 2 and not 12. Bonnie looks at the problem again. She asks the student to go to the board and to explain her strategy. S2 draws a circle and places 2 lines around the circle. He counts starting from the line he identifies as Karen up to 12 going around the circle 6 times. Bonnie looks surprised. She is looking at me for assurance. What do you think class? Which do you think is right? S3 suggests that his answer is different from the others and that he thinks there are 3 people at the party. Students are quiet. Bonnie asks S3 to show his work. S3 using the same diagram S2 has used, suggests adding another line around the circle representing another guest and shows by the means of counting and going around the circle 4 times why he thinks the correct answer is 3. Do you see what I have done? he asks Bonnie. Bonnie nods her head in approval. S4 goes back to his chair. Bonnie looks puzzled and continues to read the problem she assigned. A few seconds of silence passes as students look at Bonnie to tell them what to do. In the meantime, S4 asks if his answer, 6, could be right. He, too, is asked to go to the board and to show his solution. She specifically asks students to pay attention to S4's argument. S4 extends other students arguments by adding 2 more lines around the circle. In the meantime S5 and S6 simultaneously ask Bonnie why their answer of 12 could not be correct. Long pause by Bonnie as she looks at the solutions offered by the group. I think this was a really good problem. We all have different answers and they all sound right . . . (pause). This is real problem solving, you see! You all did very well, class. S3 asks which answer she should record on her paper. I don't understand why these answers all can be right. I still think mine is right! Should I write my answer or everybody else's? Just put down all the answers so you could think about it some more, Bonnie responds. All students begin recording the answers on their papers. (Field notes, December 14, 1996)
Teachers also varied in their level of knowledge of, and understanding about, how children learned mathematics, learners' cognitive needs and obstacles as they encountered different topics, and multiple representations and questioning techniques that could help children overcome those obstacles. In nearly all standards-based programs, problem situations and group activities were designed to create environments in which conflict and confrontation among students' ideas were conceived. The classroom social settings, however, did not always force the elaboration and justification of various positions and did not help students to extend ideas or systematically test opposing views. That fact resulted primarily from the teachers' lack of familiarity with appropriate questioning techniques and knowledge of multiple approaches to problems and investigations. Although the teachers listened to students' explanations and arguments, they did not always know how to build on those arguments to discuss an important topic or to help learners form generalizations relating to the procedures used in investigations. In some instances, teachers were not certain of or knowledgeable about multiple alternatives for solving problems and approaching tasks.
Among the participants, the beginning teachers with a constructivist perspective willingly admitted their inability to teach mathematics from a textbook perspective. That group, however, seemed more concerned about classroom management and organization of group activities than about how they taught course content.
Discussion The findings of this study suggest that the amount and quality of teachers' experiences, their professional knowledge base about curriculum and instruction, the contexts within which they worked, and their own personal theories on effective teaching and learning practices were strongly related to how they perceived, evaluated, and used the curriculum-based materials in their classrooms. The results support the findings of previous literature on the effect of teachers' personal beliefs on how they teach content (Ball, 1988a, 1988b, 1990; Brown & Baird, 1992; Brown & Borko, 1992; Brown & Cooney, 1985; Bush, 1986). Our findings substantiate the impact of content and teachers' pedagogical content knowledge in creating and sustaining instruction that promotes student discourse within the learning environment and also in teaching for conceptual understanding (Ball, 1988a, 1988b, 1990; Carlson, 1988; Manouchehri & Enderson, in press; Shulman, 1986; Shulman & Grossman, 1988; Thompson et al., 1994).
The teacher participants in this study operated on two highly intertwined spheres of self and social domains while using the curriculum materials. On a personal level, an important consideration for teachers was what the programs offered that they did not already know or have. In a practical way, teachers judged and used standards-based curricula in light of the repertoire of strategies, methods, and materials they had developed or needed to develop. This expertise was closely connected to the amount and kinds of experiences they had, their personal theories on what was best for students, and their content and pedagogical knowledge base.
On a social level, the contexts within which the teachers worked significantly affected their willingness and ability to integrate innovative methods and materials into their classrooms, as well as their eagerness and ability to deal with any obstacles that arose.
Although the teachers' personal constructs and beliefs relating to teaching, learning, students' needs, and professional knowledge were an integral part of the standards-based curriculum, their personal theories were shaped by their work setting. The teachers' work setting served as a learning environment for ideas and perspectives about teaching.
As evident in this study, personal theories are subject to change and refinement. For those teachers with strong ties to traditional curriculum and teaching, exposure to new ideas created feelings of inadequacy, especially if the teachers worked in environments in which constructivist perspectives on learning and teaching were emphasized. Similarly, those teachers with strong ties to student-centered teaching practices and constructivist theories who were placed in an environment where such practices were not supported by a larger community, were sometimes repressed.
For participating teachers in this study, changes in their beliefs and practices did not occur merely by placing new programs at their disposal, but also by initiating and guiding the development of their pedagogical understanding. Standards-based curriculum programs are only as promising as the teacher's ability to realize engaging and demanding interactions for the learning and teaching processes occurring in their classrooms. In order for such curricula to potentially change school practice, researchers must examine the needs of teachers as they relate to their content and pedagogical content growth and development. New curricula should provide all teachers with the opportunity to grow mathematically and to consider alternative representations of the ideas and connections among them. Plans for the standards-based curriculum materials derive in part from research-based knowledge of how students learn mathematics and how it can be taught effectively. The view that learning is an active, social, and constructive process, as emphasized in the philosophy of current materials, applies as much to teachers as to students.
We hope that students will be motivated to resolve their mathematics problems; teachers must have reason and motivation to recognize their pedagogical practice. Moreover, we hope that students will be enculturated in new demands envisioned by the current reform and portrayed in current textbooks; teachers must be carefully oriented to the potential of the materials by studying how their students learn and interpret their responses. Curriculum materials need to include not only exciting and mathematically significant context for student explorations but also information on how to incorporate a mechanism for systematically nurturing teachers' professional growth. For the teacher participants in this study, the programs provided little opportunity to view alternative representations of the ideas and any connections among them. The underlying assumptions were that the teachers will know the programs and be able to implement them, adopt new methods of instruction and assessment suggested by textbooks, interpret children's thinking, and design subsequent instruction. More in-depth assistance is needed for teachers to think about content, possibilities for connections, and pedagogical practices conducive to effective implementation.
The implications of this study for the mathematics education community are at least twofold. This study highlights the knowledge that reform in teaching and learning mathematics is neither simple nor straightforward. Mathematics reform must overcome the fact that many teachers do not work in risk-free environments. Moreover, many teachers do not possess a risk-free orientation toward their own teaching and learning. According to our findings in this study, systematic integration of new instruction techniques and the use of new curricular materials that impose a drastically different structure on learning environments require that the teacher enters a very uncertain world. Current knowledge about teaching and learning mathematics from a conceptual perspective is such that the teacher must be a researcher of teaching and learning. The teacher must create and test solutions to education problems and research curriculum materials--an area that is filled with ambiguities. Accumulating new knowledge about learners, mathematics, and pedagogical maps to implement authentic instruction is an ultimate form of dealing with the ambiguities. That method implies that the reform movement in mathematics education should also affect the processes used in teacher education. Reform in the teaching of mathematics, with the underlying need to empower teachers and students, must accompany changes in teacher preparation and development. If one of the goals of pedagogical practice is to reach all students, then teacher education must focus on empowering teachers to implement the mandates of the reform.
Conceptualizing change in school settings requires an understanding of how teachers' beliefs and theories effect how they teach and evaluate content; those personal theories are not formed in isolation and are not static. The theories, which are dynamic and interactive, are shaped in a socially constructed, complex, and institutionalized school setting. As a result, teachers could not begin to practice, or continue to practice, without some knowledge of the context of their practice and ideas about what can and should be done in those circumstances. Although recent research has guided our understanding of the nature of teachers' beliefs and how they influence their practice, researchers will need to investigate in greater depth the long-range impact of such social influences on teachers' practices and on student learning.
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- rodaniel Oct 23, 2009- rodaniel Oct 23, 2009
By AZITA MANOUCHEHRI, Maryville University--St. Louis and TERRY GOODMAN, Central Missouri State University
Address correspondence to Azita Manouchehri, Mathematics Education, Maryville University--St. Louis, 13550 Conway Road, St. Louis, MO 63141.

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